SCHWEITZER 1996

Let X be a $\kappa$ weighted compact $T_2$ space. Prove that for every $\omega\leq\lambda<\kappa$ , X has a continuous image of a $T_2$ space of weight $\lambda$ . (The weight of a space X is the smallest infinite cardinality of a base of X.)

A complete graph is in a plane such that no three of its vertices are collinear. The edges of the graph, which are straight segments connecting the vertices, are colored with two colors. Prove that there is a non-self-intersecting spanning tree consisting of edges of the same color.

Let $1\leq a_1 < a_2 <... < a_{2n} \leq 4n-2$ be integers, such that their sum is even. Prove that for all sufficiently large n, there exist $\varepsilon_1 , ..., \varepsilon_{2n} = \pm1$ such that $$\sum\varepsilon_i = \sum\varepsilon_i a_i = 0$$

Prove that in a finite group G the number of subgroups with index n is at most $| G |^{2 \log_2 n}$ .

Let K and D be the set of convergent and divergent series of positive terms respectively. Does there exist a bijection between K and D such that for all $\sum a_n,\sum b_n\in K$ and $\sum a_n',\sum b_n'\in D$ , $\frac{a_n}{b_n}\to 0\iff \frac{a_n'}{b_n'}\to 0$ ? Under the bijection, $\sum a_n\leftrightarrow\sum a_n'$ and $\sum b_n\leftrightarrow\sum b_n'$ .

Let $\{a_n\}$ be a bounded real sequence. (a) Prove that if X is a positive-measure subset of $\mathbb R$ , then for almost all $x\in X$ , there exist a subsequence $\{y_n\}$ of X such that $$\sum_{n=1}^\infty (n(y_n-x)-a_n)=1$$ (b) construct an unbounded sequence $\{a_n\}$ for which the above equation is also true.

Construct a holomorphic function $f(z) = \sum \limits_{n = 0} ^ \infty a_n z^n$ ( | z | <1 ) in the unit circle that can be analytically continued to all points of the unit circle except one point, and for which the sequence $\{a_n\}$ has two limit points, $\infty$ and a finite value.

Prove that a simply connected, closed manifold (i.e., compact, no boundary) cannot contain a closed, smooth submanifold of codimension 1, with odd Euler characteristic.

Let $Y_1 , ..., Y_n$ be exchangeable random variables, ie for all permutations $\pi$ , the distribution of $(Y_{\pi (1)}, \dots, Y_{\pi (n)} )$ is equal to the distribution of $(Y_1 , ..., Y_n)$ . Let $S_0 = 0$ and $$S_j = \sum_{i = 1}^j Y_i j = 1,\dots,n$$ Denote $S_{(0)} , ..., S_{(n)}$ by the ordered statistics formed by the random variables $S_0 , ..., S_n$ . Show that the distribution of $S_{(j)}$ is equal to the distribution of $\max_{0 \le i \le j} S_i + \min_ {0 \le i \le n-j} (S_{j + i} -S_j)$ .